task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | Given Isabella's house has 4 bedrooms, each with a length of 15 feet, a width of 12 feet, a height of 9 feet, and doorways and windows occupying 80 square feet, calculate the total square feet of walls that must be painted. | 1624 | 59 | 4 |
math | Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:
\begin{enumerate}
\item $\frac{a}{2} \le x \le 2a$
\item $\frac{a}{2} \le y \le 2a$
\item $x+y \ge a$
\item $x+a \ge y$
\item $y+a \ge x$
\end{enumerate}
T... | 6 | 122 | 1 |
math | Given that point $P$ is a moving point on the parabola $y^{2}=2x$, find the minimum value of the sum of the distance from point $P$ to point $(0,2)$ and the distance from $P$ to the directrix of the parabola. | \dfrac { \sqrt {17}}{2} | 62 | 12 |
math | Let $a,b,c,d$ be distinct digits such that the product of the $2$ -digit numbers $\overline{ab}$ and $\overline{cb}$ is of the form $\overline{ddd}$ . Find all possible values of $a+b+c+d$ . | 21 | 67 | 2 |
math | The wool shear yield of an Askanian breed sheep is a random variable \( X \) distributed according to a normal law. Almost all possible values of this variable belong to the interval (7; 10.6) kg. Find the interval, symmetric with respect to the mathematical expectation, in which the possible values of the wool shear y... | (7.624, 9.976) | 87 | 14 |
math | In a regular decagon $ABCDEFGHIJ$, points $K$, $L$, $M$, $N$, $O$, $P$, $Q$, $R$, and $S$ are selected on the sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, $\overline{EF}$, $\overline{FG}$, $\overline{GH}$, $\overline{HI}$, and $\overline{IJ}$ respectively. Each of these points divides their... | \frac{3\sqrt{3}}{40} | 262 | 13 |
math | A three-digit decimal rounded to two decimal places is approximately 3.58. The smallest this three-digit decimal can be is \_\_\_\_\_\_, and the largest it can be is \_\_\_\_\_\_. | 3.575, 3.584 | 47 | 12 |
math | The solution of $\sqrt{5x-1}+\sqrt{x-1}=2$ is $x=. | x=1 | 23 | 3 |
math | The solution to the fractional equation about $x$, $\frac{{3x-a}}{{x-3}}+\frac{{x+1}}{{3-x}}=1$, is a positive number. The solution set of the inequality system about $y$, $\left\{\begin{array}{l}{y+9≤2(y+2)}\\{\frac{{2y-a}}{3}＞1}\end{array}\right.$, is $y\geqslant 5$. Find the sum of all integers $a$ that satisfy the ... | 13 | 115 | 2 |
math | Let the function $f(x)$ have a derivative $f'(x)$ on $\mathbb{R}$, and for any real number $x$, it satisfies $f(x)=6x^{2}-f(-x)$. When $x\in(-\infty,0)$, $2f'(x)+1 < 12x$. If $f(m+2)\leqslant f(-2m)+12m+12-9m^{2}$, determine the range of the real number $m$. | [-\dfrac{2}{3}, +\infty) | 112 | 14 |
math | Given that Mr. and Mrs. Epsilon want to name their baby Alex such that his monogram (first, middle, and last initials) will be in alphabetical order, consists only of vowels, and the last initial is 'X', find how many such monograms are possible. | 10 | 57 | 2 |
math | A tetrahedron has a triangular base with sides all equal to 2, and each of its three lateral faces are squares. A smaller tetrahedron is placed within the larger one so that its base is parallel to the base of the larger tetrahedron and its vertices touch the midpoints of the lateral faces of the larger tetrahedron. Ca... | \frac{\sqrt{2}}{12} | 85 | 11 |
math | Given a sequence $\{a_n\}$ satisfying $a_1 = 2$, and the sum of the first $n$ terms is $S_n$.
We have the recurrence relationship for $n \geq 1$,
$$
a_{n+1}=
\begin{cases}
pa_{n}+n-1 & \text{if }n\text{ is odd} \\
-a_{n}-2n & \text{if }n\text{ is even}
\end{cases}
$$
(Ⅰ) If the sequence $\{b_n\}$ satisfies $b_n = a_{... | n = 3 | 296 | 4 |
math | Given vectors $\overrightarrow {a}=(2\cos x, \sqrt {3}\sin x)$ and $\overrightarrow {b}=(\cos x, 2\cos x)$, and a function $f(x)= \overrightarrow {a}\cdot \overrightarrow {b}+m$, where $m \in \mathbb{R}$. The minimum value of $f(x)$ when $x \in [0, \frac{\pi}{2}]$ is 2.
1. Find the interval where $f(x)$ is monotonical... | \frac{\pi}{12} + \frac{\pi}{4} + \frac{7\pi}{12} + \frac{3\pi}{4} = \frac{5\pi}{3} | 208 | 47 |
math | A certain residential community advocates a low-carbon lifestyle and encourages environmentally friendly travel by providing bicycle rental services. The community has 40 bicycles available for residents to rent, with a daily management cost of 92 yuan. Based on experience, if the daily rental price of each bicycle doe... | 220 \text{ yuan} | 178 | 8 |
math | Given the sequence $1990-1980+1970-1960+\cdots -20+10$, calculate the sum. | 1000 | 37 | 4 |
math | A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment ... | 4 | 81 | 1 |
math | Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function.
*Proposed by David Altizio* | 50 | 59 | 2 |
math | Let $n$ be the smallest positive integer that is a multiple of 75 and has exactly 100 positive integral divisors, including 1 and itself. Find $\frac{n}{75}$. | 3600 | 44 | 4 |
math | Consider polynomials of the form $x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + 2023$, where $a$, $b$, $c$, $d$, and $e$ are real numbers. How many such polynomials exist such that if $s$ is a root, so is $\frac{-1-i\sqrt{3}}{2} \cdot s$? | 1 | 92 | 1 |
math | A large candle is $150$ centimeters tall. It burns down at varying rates, taking $15$ seconds to burn the first centimeter, $30$ seconds for the second centimeter, and $15k$ seconds for the $k$-th centimeter. Calculate the height of the candle $\tfrac{T}{2}$ seconds after it is lit, where $T$ is the total time for the ... | 44 \text{ centimeters} | 96 | 8 |
math | Find the mass of the plate $D$ with surface density $\mu = \frac{x^2}{x^2 + y^2}$, bounded by the curves
$$
y^2 - 4y + x^2 = 0, \quad y^2 - 8y + x^2 = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0.
$$ | \pi + \frac{3\sqrt{3}}{8} | 90 | 15 |
math | Given an ellipse M: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ (a>0, b>0) with two vertices A(-a, 0) and B(a, 0). Point P is a point on the ellipse distinct from A and B. The slopes of lines PA and PB are k₁ and k₂, respectively, and $$k_{1}k_{2}=- \frac {1}{2}$$.
(1) Find the eccentricity of the ellipse C.
(2) I... | \frac { \sqrt {2}}{2} | 165 | 11 |
math | A rectangular park is one-third as wide as it is long, and it is completely enclosed by 90 meters of fencing. What is the number of square meters in the area of the park? | 379.6875 | 40 | 8 |
math | Three of the four vertices of a rectangle are at $(4, 10)$, $(15, 10)$, and $(15, -3)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 4)^2 + (y + 3)^2 = 16$? Express your answer in terms of $\pi$. | 4\pi | 86 | 3 |
math | Given $f\left(x\right)=\frac{{x}^{2}}{1+{x}^{2}}$, if $f(\sqrt{1})=\frac{1}{2}$, $f(\sqrt{\frac{1}{2}})=\frac{1}{3}$, then find the value of $f(\sqrt{1})+f(\sqrt{2})+f(\sqrt{\frac{1}{2}})+f(\sqrt{3})+f(\sqrt{\frac{1}{3}})+…+f(\sqrt{n})+f(\sqrt{\frac{1}{n}})$ | n - \frac{1}{2} | 131 | 9 |
math | Given that the random variable $X$ follows a normal distribution with mean $1$ and variance $\sigma ^{2}$, and $P(X\leq 4)=0.78$, calculate the probability $P(X<-2)$. | 0.22 | 51 | 4 |
math | Given the function $f(x)=\sin (\sqrt{3}x+φ)$ with $0 < φ < π$, if $f(x)+f{{'}}(x)$ is an odd function, find the value of $φ$. | \dfrac{2π}{3} | 49 | 8 |
math | Each of the integers 1 to 7 is to be written, one in each circle in the diagram, and the sum of the three integers in any straight line is to be the same. Calculate the number of different ways the centre circle can be filled. | 3 | 52 | 1 |
math | Given two lines $l_1$: $(m+3)x+4y=5-3m$ and $l_2$: $2x+(m+5)y=8$. For what values of $m$ do the following conditions hold: (1) $l_1$ is parallel to $l_2$; (2) $l_1$ coincides with $l_2$; (3) $l_1$ is perpendicular to $l_2$. | m=-\frac{13}{3} | 102 | 10 |
math | $ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$ . What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$ ?
*2018 CCA Math Bonanza Lightning Round #1.3* | 45^\circ | 81 | 4 |
math | The rectangular flag shown is divided into seven stripes of equal height. The height of the flag is $h$ and the length of the flag is twice its height. The total area of the four shaded regions is $1400 \mathrm{~cm}^{2}$. What is the height of the flag? | 35 \mathrm{~cm} | 65 | 8 |
math | In parallelogram \(ABCD\), the side \(AD\) is divided into equal parts by points \(A_1, A_2, \ldots, A_{2022}\). Point \(E_1\) is the intersection of lines \(BA_1\) and \(AC\). Determine what fraction of the diagonal \(AC\) is the segment \(AE_1\). | \frac{1}{2024} | 81 | 10 |
math | Given a $4 \times 4$ grid with 16 unit squares, each painted white or black independently and with equal probability, find the probability that the entire grid becomes black after a 90° clockwise rotation, where any white square landing on a place previously occupied by a black square is repainted black. | \frac{1}{65536} | 65 | 11 |
math | Three distinct vertices of a tetrahedron are chosen at random. Determine the probability that the plane determined by these three vertices contains points inside the tetrahedron. | 0 | 34 | 1 |
math | Given the parabola $C$: $y^{2}=4x$ with focus $F$ and a point $P(1,m)$ on parabola $C$.
(1) If the ellipse $C'$: $\frac{x^{2}}{4}+ \frac{y^{2}}{n}=1$ shares a common focus with the parabola $C$, find the equation of ellipse $C'$.
(2) Let $A$ and $B$ be the intersection points of the parabola $C$ and the ellipse $C'$ fo... | 3x^{2}- \frac{y^{2}}{2}=1 | 159 | 16 |
math | Determine a base in which the number $729_{10}$ is expressed having the form XYXY, where X and Y are distinct digits. Identify the base. | 8 | 36 | 1 |
math | $M$ is an $8 \times 8$ matrix. For $1 \leq i \leq 8$, all entries in row $i$ are at least $i$, and all entries on column $i$ are at least $i$. What is the minimum possible sum of the entries of $M$ ? | 372 | 68 | 3 |
math | Given James writes down fifteen 1's in a row and randomly writes $+$ or $-$ between each pair of consecutive 1's, calculate the probability that the value of the expression James wrote down is $7$. | \frac{364}{2^{14}} | 49 | 12 |
math | Consider six-digit palindromes where the digits form the pattern $abccba$. Compute the sum of all such six-digit palindromes, and then find the sum of the digits of this total sum. | 45 | 44 | 2 |
math | Initially, the number 1 and two positive numbers \( x \) and \( y \) are written on a blackboard. In each move, a player can choose any two numbers on the board, not necessarily distinct, and write their sum or their difference on the board. Additionally, they can choose any non-zero number on the board and write its i... | xy | 105 | 1 |
math | Find the equation of the tangent line to the curve $y=\sin x+e^{x}$ at $x=0$.
A) $x-3y+3=0$
B) $x-2y+2=0$
C) $2x-y+1=0$
D) $3x-y+1=0$ | 2x - y + 1 = 0 | 73 | 10 |
math | In a triangle with base $b = 10$ and height $h = 12$, a circle is inscribed touching the base of the triangle at a single point and the other two sides tangentially. Find the radius $r$ of the circle. | \frac{10}{3} | 54 | 8 |
math | Let \(a\), \(b\), and \(c\) be positive real numbers such that \(a + 2b + 3c = 1.\) Find the minimum value of
\[
\frac{1}{a} + \frac{2}{b} + \frac{3}{c}.
\] | 36 | 67 | 2 |
math | Solve the equations using the specified methods:<br/>
(1) $x^{2}-4x-2=0$ (using factoring method);<br/>
(2) $2y^{2}-3y-1=0$ (using formula method);<br/>
(3) $3x(x-1)=2-2x$ (using appropriate method);<br/>
(4) $2x^{2}-x-1=0$ (using factoring method). | x_{1}=1, x_{2}=-\frac{1}{2} | 103 | 18 |
math | In a new diagram, $\overrightarrow{OA}\perp\overrightarrow{OC}$ and $\overrightarrow{OB}\perp\overrightarrow{OD}$. If angle $\angle{AOD}$ is 2.5 times $\angle{BOC}$, what is $\angle{AOD}$? | 128.57^\circ | 65 | 8 |
math | A reconnaissance squad has 12 soldiers, including 3 radio operators. These 12 soldiers are to be randomly divided into 3 groups, with 3 soldiers, 4 soldiers, and 5 soldiers in each group. What is the probability that each group includes exactly 1 radio operator? | \frac{3}{11} | 61 | 8 |
math | In the elective course 4-4: Coordinate System and Parametric Equations, the line described by the polar equation $\rho\cos\theta-\rho\sin\theta-1=0$ intersects the x-axis at point P, and intersects the ellipse described by the parametric equations $\begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$ (where $\theta... | \frac{6}{5} | 106 | 7 |
math | Given a non-negative sequence $\left\{a_{n}\right\}(n \geqslant 1)$ with the first term $a$, and the sum of the first $n$ terms $S_{n}$, where $S_{n}=\left(\sqrt{S_{n-1}}+\sqrt{a}\right)^{2}$ for $n \geqslant 2$. If $b_{n}=\frac{a_{n+1}}{a_{n}}+\frac{a_{n}}{a_{n+1}}$, find the sum of the first $n$ terms of the sequence... | \frac{4n^2 + 6n}{2n + 1} | 155 | 18 |
math | Define the minimum distance from a point on curve C to line l as the distance between curve C and line l. Given that the distance from curve $C_1: y = x^2 + a$ to line $l: y = x$ is equal to the distance from curve $C_2: x^2 + (y+4)^2 = 2$ to line $l: y = x$, find the value of the real number $a$. | a = \frac{9}{4} | 96 | 9 |
math | Given the complex number $\dfrac {2-i}{1-i}$, simplify this expression. | \dfrac {3}{2}+ \dfrac {i}{2} | 19 | 16 |
math | Given two circles \\({{F}_{1}}:{{\left( x+\sqrt{3} \right)}^{2}}+{{y}^{2}}=9\\) and \\({{F}_{2}}:{{\left( x-\sqrt{3} \right)}^{2}}+{{y}^{2}}=1\\), and an ellipse \\(C:\dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1 \left( a > b > 0 \right)\\) with the centers of circles \\({F}_{1},{F}_{2} \\) as its foci, which pas... | 3 | 294 | 1 |
math | Triangle $OPQ$ has vertices $O=(0,0)$, $P=(4,0)$, and $Q$ in the first quadrant. The angle $PQO$ is $90^\circ$ and the angle $POQ$ is $45^\circ$. Suppose that $OQ$ is rotated $45^\circ$ clockwise about $O$. Find the coordinates of the image of $Q$. | (4\sqrt{2}, 0) | 89 | 10 |
math | Let $A=\{x|y=\sqrt{3-x}+\log_{2}(\sqrt{x+2}+1)\}$, $B=\{x|2-m\leq x\leq 2m-3\}$.
$(1)$ If $p:x\in A$, $q:x\in B$, and $p$ is a sufficient but not necessary condition for $q$, find the range of real number $m$.
$(2)$ If $A\cup B=A$, find the range of real number $m$. | (-\infty, 3] | 117 | 8 |
math | Given the function $f(x)=a\ln x$ ($a\in \mathbb{R}$).
- (I) If the function $g(x)=2x+f(x)$ has a minimum value of $0$, find the value of $a$;
- (II) Let $h(x)=f(x)+ax^{2}+(a^{2}+2)x$, find the monotonic intervals of the function $h(x)$;
- (III) Suppose the function $y=f(x)$ and the function $u(x)= \frac {x-1}{2x}$ have... | \frac {1}{2} | 161 | 7 |
math | Given the sequence $\{a_n\}$ with the general term formula $a_n = -n^2 + 12n - 32$, determine the maximum value of $S_n - S_m$ for any $m, n \in \mathbb{N^*}$ and $m < n$. | 10 | 66 | 2 |
math | What is the smallest number of participants that can be in a math club, given that the number of girls is less than 50% but more than 40%? | 7 | 36 | 1 |
math | Evaluate \(99 \times 105\) in your head. | 10395 | 15 | 5 |
math | Given two circles $C_1: {(x-1)^2}+{(y-2)^2}=4$ and $C_2: {(x-2)^2}+{(y-1)^2}=2$ intersect at points $A$ and $B$. Line $l$ is parallel to line $AB$, tangent to circle $C_{2}$, and intersects circles $C_{1}$ and $C_{2}$ at points $M$ and $N$ respectively. Find the length of $|MN|$. | 4 | 112 | 1 |
math | How many numbers can be formed using the digits 0, 1, 2, 3, 4 to create:
(1) A four-digit number?
(2) A four-digit even number?
(3) A four-digit number without repeating digits?
(4) A four-digit even number without repeating digits? | 60 | 70 | 2 |
math | Rebecca has four resistors, each with resistance 1 ohm . Every minute, she chooses any two resistors with resistance of $a$ and $b$ ohms respectively, and combine them into one by one of the following methods: - Connect them in series, which produces a resistor with resistance of $a+b$ ohms; - Connect them in parallel,... | 15 | 154 | 2 |
math | There are 70 chips in a box. Each chip is either red or blue. If the sum of the number of red chips and twice the number of blue chips equals a prime number, what is the least possible number of red chips? | 69 | 49 | 2 |
math | It takes Jana 24 minutes to walk one mile. If she includes a 6-minute rest after every mile walked, how far will she walk in 78 minutes? Express your answer as a decimal to the nearest tenth. | 2.0 \text{ miles} | 48 | 8 |
math | What is the largest number of integers that we can choose from the set $\{1, 2, 3, \ldots, 2017\}$ such that the difference between any two of them is not a prime number? | 505 | 50 | 3 |
math | If the complex numbers $z_{1}$ and $z_{2}$ correspond to points on the complex plane that are symmetric about the y-axis, and $z_{1}=2-i$, find the quadrant of the complex plane where the point corresponding to the complex number $\frac{z_{1}}{z_{2}}$ is located. | 2 | 70 | 1 |
math | The pattern on a large square tile consists of eight congruent right-angled triangles and a small square. The area of the tile is \(49 \, \text{cm}^2\) and the length of the hypotenuse \(PQ\) of one of the triangles is \(5 \, \text{cm}\). What is the area of the small square?
A) \(1 \, \text{cm}^2\)
B) \(4 \, \text{c... | 1 \, \text{cm}^2 | 153 | 10 |
math |
A starship enters an extraordinary meteor shower. Some of the meteors travel along a straight line at the same speed, equally spaced. Another group of meteors travels similarly along another straight line, parallel to the first, with the same speed but in the opposite direction, also equally spaced. The ship travels p... | 9.1 | 171 | 3 |
math | After the first "H expansion", the sequence becomes 1, 3, 2; after the second "H expansion", it becomes 1, 4, 3, 5, 2. Find the number of items in the sequence after the 10th "H expansion". | 1025 | 61 | 4 |
math | The number of nonzero complex numbers $z$ that satisfy the condition that the points $0, z, z^4$ in the complex plane are vertices of a square. | 4 | 35 | 1 |
math | The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is | 17 | 28 | 2 |
math | Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square.
Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers. | n | 75 | 2 |
math | Given the following four propositions:<br/>① If a line is perpendicular to two lines in a plane, then the line is perpendicular to the plane;<br/>② If a line is perpendicular to any line in a plane, then the line is perpendicular to the plane;<br/>③ If a line is perpendicular to the lines containing the two legs of a t... | 2 | 150 | 1 |
math | Write the number \( 123456789101112 \cdots 19941995 \) on the blackboard to form an integer \( N_1 \). Erase the digits in the even positions of \( N_1 \) to form \( N_2 \). Then, erase the digits in the odd positions of \( N_2 \) to form \( N_3 \). Continue this process of erasing digits in even and odd positions alte... | 9 | 186 | 1 |
math | Given the ellipse defined by the equation $2x^2 + y^2 = 8$, determine the coordinates of the foci. | (0, \pm 2) | 28 | 8 |
math | Шоссе Долгое пересекается с улицей Узкой и с улицей Тихой. На обоих перекрёстках стоят светофоры. Первый светофор $x$ секунд разрешает движение по шоссе, а полминуты - по ул. Узкой. Второй светофор две минуты разрешает движение по шоссе, а $x$ секунд - по ул. Тихой. Светофоры работают независимо друг от друга. При како... | \frac{4}{9} | 192 | 7 |
math | Thirty gremlins and twenty imps attend the Annual Mischief Convention. Due to some alliances, exactly five imps are willing to shake hands with each other but refuse to shake hands with the remaining imps. All imps shake hands with all gremlins. Meanwhile, being sociable, all gremlins shake hands with each other as wel... | 1045 | 90 | 4 |
math | In the geometric sequence $\{a\_n\}$, if $a\_1=-1$, $a\_2+a\_3=-2$, then its common ratio is $\_\_\_\_\_\_$. | -2 \text{ or } 1 | 43 | 9 |
math | The angle $A$ at the vertex of the isosceles triangle $ABC$ is $100^{\circ}$. On the ray $AB$, a segment $AM$ is laid off, equal to the base $BC$. Find the measure of the angle $BCM$. | 10 | 59 | 2 |
math | Calculate the area of a right-angled trapezoid, if its acute angle is $60^{\circ}$, the smaller base is $a$, and the longer leg is $b$. | \frac{(2a + b)b \sqrt{3}}{4} | 41 | 16 |
math | Machine tools A, B, and C each independently process the same type of part. It is known that the probabilities of the parts processed by machine tools A, B, and C being first-class are 0.7, 0.6, and 0.8, respectively. The number of parts processed by machine tools B and C are equal, and the number of parts processed by... | 0.6517 | 202 | 6 |
math | Given real numbers $x$ and $y$ satisfy that three of the four numbers $x+y$, $x-y$, $\frac{x}{y}$, and $xy$ are equal, determine the value of $|y|-|x|$. | \frac{1}{2} | 51 | 7 |
math | There exist two distinct unit vectors $\mathbf{v}$ such that the angle between $\mathbf{v}$ and $\begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix}$ is $45^\circ,$ and the angle between $\mathbf{v}$ and $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$ is $60^\circ.$ Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be these vectors. Find $\... | \sqrt{2} | 133 | 5 |
math | In a quadrilateral pyramid \(S A B C D\):
- The lateral faces \(S A B, S B C, S C D, S D A\) have areas of 9, 9, 27, and 27 respectively;
- The dihedral angles at the edges \(A B, B C, C D, D A\) are equal;
- The quadrilateral \(A B C D\) is inscribed in a circle, and its area is 36.
Find the volume of the pyramid \(... | 54 | 117 | 2 |
math | A math conference is expanding its lecture series and will now feature seven different lecturers. If Dr. Smith's lecture depends on Dr. Jones's lecture, and Dr. Lee's lecture depends on both Dr. Jones's and Dr. Smith's lecture, so that Dr. Smith must be scheduled at some time after Dr. Jones and Dr. Lee must be schedul... | 120 | 95 | 3 |
math | From head to tail of the zebra Hippotigris, there are 360 stripes of equal width. Fleas Masha and Dasha started crawling from the head to the tail of the zebra. At the same time, flea Sasha started crawling from the tail to the head. Flea Dasha crawls twice as fast as flea Masha. Before meeting flea Sasha, Masha covere... | 240 \text{ stripes} | 101 | 8 |
math | Compute the sum of the series:
\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))) \] | 126 | 39 | 3 |
math | Given an arithmetic sequence $\{a\_n\}$ with a common ratio $q > 1$, $a\_1 + a\_4 = 9$, and $a\_2 a\_3 = 8$.
(1) Find the general term formula for the sequence;
(2) If $b\_n = a\_{n+1} \log \_2 a\_{n+1}$, and $S\_n$ denotes the sum of the first $n$ terms of the sequence $\{b\_n\}$, find the smallest positive integer $n... | 5 | 144 | 1 |
math | Given circle $O$ with diameter $\overline{AB}$, and points $C$, $D$, and $P$ on the same side of $\overline{AB}$, calculate the ratio of the area of the smaller sector $CPD$ to the area of the circle given $\angle AOC = 40^{\circ}$, $\angle DOB = 60^{\circ}$, and $\angle COP = 110^{\circ}$. | \frac{1}{36} | 99 | 8 |
math | Let
\[ f(x) = x^3 + 4x^2 + 13x + 20. \]
The graphs of $y = f(x)$ and $y = f^{-1}(x)$ intersect at exactly one point $(a,b)$. Enter the ordered pair $(a,b)$. | (-2, -2) | 66 | 6 |
math | The method for finding the area of a triangle given the lengths of its three sides, as described in the "Nine Chapters on the Mathematical Art," filled a gap in traditional Chinese mathematics. This method is completely equivalent to Heron's formula, demonstrating the high level of mathematics in ancient China. The met... | \sqrt{3} | 252 | 5 |
math | In triangle \( ABC \), the median \( AM \) (the segment connecting vertex \( A \) with the midpoint \( M \) of side \( BC \)) is extended beyond point \( M \) by a length equal to \( AM \). Find the distance from the resulting point to points \( B \) and \( C \) if the sides \( AB \) and \( AC \) are 5 and 4, respectiv... | NB = 4, \; NC = 5 | 90 | 11 |
math | Describe all positive integer solutions $(m, n)$ of the equation $8m - 7 = n^2$ and provide the first value of $m$ (if it exists) greater than 1959. | 2017 | 46 | 4 |
math | Given $\overline{AD} \| \overline{FG}$, find the measure of angle $EFG$.
[asy]
import olympiad;
pair A = (-15,20);
pair B = (-12,35);
pair C = (35,50);
pair D = (35,20);
pair E = (14,20);
pair F = (0,0);
pair G = (40,0);
draw(F--G);
draw(F--C);
draw(A--D);
draw(B--E);
label("F", F, W);
label("G", G, ENE);
label("C"... | 60^\circ | 278 | 4 |
math | Given $x - y > x$ and $3x + 2y < 2y$, determine the relationship between $x$ and $y$. | x < 0, y < 0 | 32 | 9 |
math | In the set of numbers 1, 2, 3, 4, 5, select an even number a and an odd number b to form a vector $\overrightarrow{a} = (a, b)$ with the origin as the starting point. From all the vectors obtained with the origin as the starting point, select any two vectors as adjacent sides to form a parallelogram. Let the total numb... | \frac{1}{3} | 126 | 7 |
math | Tom Sawyer needs to paint a very long fence such that any two boards spaced exactly two, three, or five boards apart from one another must be painted different colors. What is the minimum number of colors Tom requires for this task? | 3 | 46 | 1 |
math | Given the function $f(x) = |2x+1| + |3x-2|$, and the solution set of the inequality $f(x) \leq 5$ is $\left\{x \mid -\frac{4a}{5} \leq x \leq \frac{3b}{5}\right\}$, where $a, b \in \mathbb{R}$.
(1) Find the values of $a$ and $b$;
(2) For any real number $x$, the inequality $|x-a| + |x+b| \geq m^2 - 3m + 5$ holds, f... | 2 | 155 | 1 |
math | Twelve mayoral candidates each made a statement about how many times lies had been told before their turn. The first candidate said, "Before me, one lie was told." The second candidate said, "Now, two lies have been told." The third candidate said, "Now, three lies have been told," and so on, until the twelfth candidat... | 11 | 118 | 2 |
math | A piece of alloy weighing 6 kg contains copper. Another piece of alloy weighing 8 kg contains copper in a different percentage than the first piece. A certain part was separated from the first piece, and a part twice as heavy was separated from the second piece. Each of the separated parts was then alloyed with the rem... | 2.4 | 96 | 3 |
math | Encrypt integers using the following method: each digit of the number becomes the units digit of its product with 7, then replace each digit $a$ with $10-a$. If a number is encrypted using the above method and the result is 473392, then the original number is ______. | 891134 | 64 | 6 |
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